On the spectral radii and the signless Laplacian spectral radii of c-cyclic graphs with fixed maximum degree

AbstractIf G is a connected undirected simple graph on n vertices and n+c-1 edges, then G is called a c-cyclic graph. Specially, G is called a tricyclic graph if c=3. Let Δ(G) be the maximum degree of G. In this paper, we determine the structural characterizations of the c-cyclic graphs, which have the maximum spectral radii (resp. signless Laplacian spectral radii) in the class of c-cyclic graphs on n vertices with fixed maximum degree Δ⩾n+c+12. Moreover, we prove that the spectral radius of a tricyclic graph G strictly increases with its maximum degree when Δ(G)⩾1+6+2n32, and identify the first six largest spectral radii and the corresponding graphs in the class of tricyclic graphs on n vertices

On the maximum AαA_{\alpha}-spectral radius of unicyclic and bicyclic graphs with fixed girth or fixed number of pendant vertices

For a connected graph $G$, let $A(G)$ be the adjacency matrix of $G$ and $D(G)$ be the diagonal matrix of the degrees of the vertices in $G$. The $A_{\alpha}$-matrix of $G$ is defined as \begin{align*} A_\alpha (G) = \alpha D(G) + (1-\alpha) A(G) \quad \text{for any $\alpha \in [0,1]$}. \end{align*} The largest eigenvalue of $A_{\alpha}(G)$ is called the $A_{\alpha}$-spectral radius of $G$. In this article, we characterize the graphs with maximum $A_{\alpha}$-spectral radius among the class of unicyclic and bicyclic graphs of order $n$ with fixed girth $g$. Also, we identify the unique graphs with maximum $A_{\alpha}$-spectral radius among the class of unicyclic and bicyclic graphs of order $n$ with $k$ pendant vertices.Comment: 16 page

Q-spectral and L-spectral radius of subgroup graphs of dihedral group

Research on Q-spectral and L-spectral radius of graph has been attracted many attentions. In other hand, several graphs associated with group have been introduced. Based on the absence of research on Q-spectral and L-spectral radius of subgroup graph of dihedral group, we do this research. We compute Q-spectral and L-spectral radius of subgroup graph of dihedral group and their complement, for several normal subgroups. Q-spectrum and Lspectrum of these graphs are also observed and we conclude that all graphs we discussed in this paper are Q-integral dan L-integral

Metric dimensions of bicyclic graphs

The distance d(va, vb) between two vertices of a simple connected graph G is the length of the shortest path between va and vb. Vertices va, vb of G are considered to be resolved by a vertex v if d(va, v) 6= d(vb, v). An ordered set W = fv1, v2, v3, . . . , vsg V(G) is said to be a resolving set for G, if for any va, vb 2 V(G), 9 vi 2 W 3 d(va, vi) 6= d(vb, vi). The representation of vertex v with respect to W is denoted by r(vjW) and is an s-vector(s-tuple) (d(v, v1), d(v, v2), d(v, v3), . . . , d(v, vs)). Using representation r(vjW), we can say that W is a resolving set if, for any two vertices va, vb 2 V(G), we have r(vajW) 6= r(vbjW). A minimal resolving set is termed a metric basis for G. The cardinality of the metric basis set is called the metric dimension of G, represented by dim(G). In this article, we study the metric dimension of two types of bicyclic graphs. The obtained results prove that they have constant metric dimension